This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 2

2011 N.N. Mihăileanu Individual, 1
Tags: quadratics , algerba , Polynom , algebra
Let be a quadratic polynom that has the property that the modulus of the sum between the leading and the free coefficient is smaller than the modulus of the middle coefficient. Prove that this polynom admits two distinct real roots, one belonging to the interval $ (-1,1) , $ and the other belonging outside of the interval $ (-1,1). $

2023 Euler Olympiad, Round 2, 1
Tags: Euler , Georgia , algerba , 100 , power of 2 , residue , number theory
Consider a sequence of 100 positive integers. Each member of the sequence, starting from the second one, is derived by either multiplying the previous number by 2 or dividing it by 16. Is it possible for the sum of these 100 numbers to be equal to $2^{2023}$? [i]Proposed by Nika Glunchadze, Georgia[/i]